Long-wavelength instability of coherent structures in plane Couette flow

Origin of localized snakes-and-ladders solutions of plane Couette flow

Spatially localized invariant solutions of plane Couette flow are organized in a snakes-and-ladders structure strikingly similar to that observed for simpler pattern-forming partial differential equations [Schneider, Gibson, and Burke, Phys. Rev. Lett. 104, 104501 (2010)PRLTAO0031-900710.1103/PhysRevLett.104.104501]. We demonstrate the mechanism by which these snaking solutions originate from well-known periodic states of the Taylor-Couette system. They are formed by a localized slug of wavy-vortex flow that emerges from a background of Taylor vortices via a modulational sideband instability. This mechanism suggests a close connection between pattern-formation theory and Navier-Stokes flow.

Streamwise decay of localized states in channel flow

Channel flow, the pressure driven flow between parallel plates, has exact coherent structures that show various degrees of localization. For states which are localized in streamwise direction but extended in spanwise direction, we show that they are exponentially localized, with decay constants that are different on the upstream and downstream sides. We extend the analysis of Brand and Gibson [J. Fluid Mech. 750, R1 (2014)]JFLSA70022-112010.1017/jfm.2014.285 for stationary states to the case of advected structures that is needed here, and derive expressions for the decay in terms of eigenvalues and eigenfunctions of certain second order differential equations. The results are in very good agreement with observations on exact coherent structures of different transversal wavelengths.

Reduced description of exact coherent states in parallel shear flows

A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower- and upper-branch states are characterized in detail.

Localization in a spanwise-extended model of plane Couette flow

We consider a nine-partial-differential-equation (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localization in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Spatially localized, time-periodic solutions were found instead, which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localized states in very wide domains. This smooth localization behavior, which has also been seen in fully resolved duct flow (S. Okino, Ph.D. thesis, Kyoto University, Kyoto, 2011), indicates that an apparently global flow structure does not have to suffer a modulational instability to localize in wide domains.

Crisis bifurcations in plane Poiseuille flow

Many shear flows follow a route to turbulence that has striking similarities to bifurcation scenarios in low-dimensional dynamical systems. Among the bifurcations that appear, crisis bifurcations are important because they cause global transitions between open and closed attractors, or indicate drastic increases in the range of the state space that is covered by the dynamics. We here study exterior and interior crisis bifurcations in direct numerical simulations of transitional plane Poiseuille flow in a mirror-symmetric subspace. We trace the state space dynamics from the appearance of the first three-dimensional exact coherent structures to the transition from an attractor to a chaotic saddle in an exterior crisis. For intermediate Reynolds numbers, the attractor undergoes several interior crises, in which new states appear and intermittent behavior can be observed. The bifurcations contribute to increasing the complexity of the dynamics and to a more dense coverage of state space.